Magnetic Solenoid for Generating a Substantially Uniform Magnetic Field

ABSTRACT

One embodiment of the invention includes a magnetic solenoid. The magnetic solenoid includes an elongated sidewall that extends between spaced apart ends. The elongated sidewall can surround a central axis that extends longitudinally along the sidewall. The elongated sidewall can have a radius that is defined by a compound equation that varies the radius as a function of position along the central axis.

TECHNICAL FIELD

The present invention relates generally to magnetic device systems, andspecifically to a magnetic solenoid for generating a substantiallyuniform magnetic field.

BACKGROUND

Magnetic solenoids can be implemented for a variety of applications togenerate a magnetic field, such as in an inner volume of the magneticsolenoid. As an example, magnetic solenoids can be implemented togenerate a magnetic field for a gyroscope, such as a nuclear magneticresonance (NMR) gyroscope that is located within the inner volume of themagnetic solenoid, to induce precession of noble gas isotopes. As anexample, magnetic solenoids can be formed of a conductive coil that isconfigured with a cylindrical geometry. Depending on the application forwhich the magnetic solenoid is intended, such as in an NMR gyroscopeapplication, it may be desirable to generate a magnetic field that issubstantially uniform throughout the inner volume of the magneticsolenoid. However, a cylindrical configuration of the conductive coilcan result in an unacceptable non-uniformity of the magnetic field, suchas near the ends of the cylindrical configuration and at points that areoff-axis from a central axis of the cylindrical configuration.

SUMMARY

One embodiment of the invention includes a magnetic solenoid. Themagnetic solenoid includes an elongated sidewall that extends betweenspaced apart ends. The elongated sidewall can surround a central axisthat extends longitudinally along the sidewall. The elongated sidewallcan have a radius that is defined by a compound equation that varies theradius as a function of position along the central axis.

Another embodiment of the invention includes a magnetic solenoid. Themagnetic solenoid includes an elongated sidewall that extends betweenspaced apart ends. The elongated sidewall can surround a central axisthat extends longitudinally along the sidewall. The elongated sidewallcan have a radius that is defined by a compound equation having a firstoperand that affects along-axis uniformity of a substantially uniformmagnetic field and a second operand that affects off-axis uniformity ofthe substantially uniform magnetic field, such that the substantiallyuniform magnetic field has a substantially uniform magnitude anddirection with respect to each point in three-dimensional space withinan inner volume that is enclosed by the conductor coil.

Another embodiment of the invention includes a magnetic solenoid that isconfigured to provide a substantially uniform magnetic field in an innervolume that is enclosed by the magnetic solenoid. The magnetic solenoidincludes a central portion in which a radius of the magnetic solenoidabout a central axis is substantially elliptical. An elliptical minoraxis occupies a plane that is normal to the central axis. The magneticsolenoid also includes first and second end portions opposite each otherin which the radius of the conductor coil flares outward from thecentral axis.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example of a perspective view of a magneticsolenoid in accordance with an aspect of the invention.

FIG. 2 illustrates an example of graphs demonstrating along-axismagnetic field in accordance with an aspect of the invention.

FIG. 3 illustrates an example of graphs demonstrating off-axis magneticfield in accordance with an aspect of the invention.

FIG. 4 illustrates an example of a gyroscope system in accordance withan aspect of the invention

DETAILED DESCRIPTION

The present invention relates generally to magnetic device systems, andspecifically to a magnetic solenoid for generating a substantiallyuniform magnetic field. The magnetic solenoid can include a conductivecoil that has a radius about a central axis that is defined by acompound equation at each point along the central axis. As an example,the compound equation can have a first operand that can be an ellipticalor circular function having a minor axis that defines the radius of theconductive coil at a midpoint of the magnetic solenoid along the centralaxis. The first operand can define a rounded center portion of themagnetic solenoid having a variable radius of the conductive coil alongthe length, with the radius having a maximum value at the midpoint ofthe magnetic solenoid along the central axis. The first operand can thusbe effective to provide along-axis uniformity of the magnetic fieldwithin the inner volume of the magnetic solenoid. The compound equationcan also have a second operand that can be, for example, one of anexponential, parabolic, or hyperbolic function. The second operand candefine a flaring of the radius of the conductive coil away from thecentral axis at each end of the magnetic solenoid. The second operandcan thus be effective to provide off-axis uniformity of the magneticfield within the inner volume of the magnetic solenoid. The magneticsolenoid can thus be substantially symmetrical about a plane at themidpoint of the magnetic solenoid along the central axis.

As another example, the compound equation can have three operands. Thefirst operand can define a radius at the midpoint of the magneticsolenoid along the central axis. The second and third operands can eachbe exponential functions. The second and third operands can each includepre-selected constants and can have magnitudes that vary as a functionof distance from the midpoint of the magnetic solenoid along the centralaxis. As an example, the second operand can be subtracted from the firstoperand and the third operand can be added to the first operand.Therefore, the second and third operands can be selected to vary theradius along the central axis to define a rounded center portion of themagnetic solenoid having a variable radius of the conductive coil alongthe length and to define a flaring of the radius of the conductive coilaway from the central axis at each end of the magnetic solenoid.

FIG. 1 illustrates an example of a perspective view of a magneticsolenoid 10 in accordance with an aspect of the invention. As anexample, the magnetic solenoid 10 can be implemented in any of a varietyof applications that utilize a magnetic field, such as in a nuclearmagnetic resonance (NMR) gyroscope application. In the example of FIG.1, the magnetic solenoid 10 is demonstrated in a side view 12 and in anisometric view 14. The magnetic solenoid 10 includes a coil form 16,around which is wound a conductive coil (not shown) that is configuredto generate a substantially uniform magnetic field within an innervolume 18 of the magnetic solenoid 10. As an example, the coil form 16can be encased in a magnetic shielding material (not shown) which servesto improve the field uniformity. For example, the magnetic shieldingmaterial can decrease the field fluctuations caused by external fields,such as the natural magnetic field of Earth or from artificial sources.As another example, a well designed and constructed high-permeabilitymagnetic shield can, through reflective image fields, significantlyenhance the uniformity of the magnetic field generated by the magneticsolenoid 10. The conductive coil can have a radius about a central axis20 that conforms to an outer-diameter (OD) of the coil form 16.Therefore, the magnetic field that is generated within the inner volume18 has a magnitude that is substantially uniform at each point withinthe inner volume 18 based on the radius of the conductive coil aroundthe coil form 16.

The radius of the conductive coil can be defined by a compound equationat each point along the central axis 20 within the inner volume 18 ofthe magnetic solenoid 10. As described herein, a compound equation isdefined as an equation having a solution that is defined by two or moreoperands that each includes at least one variable. Thus, because thecompound equation defines the radius at each point along the centralaxis 20 within the inner volume 18 of the magnetic solenoid 10, eachoperand of the compound equation includes a variable that is thelocation of each point along the central axis 20 within the inner volume18 of the magnetic solenoid 10.

As an example, a first operand of the compound equation can be acircular or an elliptical function having a minor axis that defines theradius of the conductive coil at a midpoint of the magnetic solenoidalong the central axis 20. A second operand of the compound equation canbe an exponential, parabolic, or hyperbolic function that defines aflaring of the radius of the conductive coil away from the central axis20 at each end of the magnetic solenoid 10. As another example, a firstoperand of the compound equation can define a radius at the midpoint ofthe magnetic solenoid 10 along the central axis 20. A second and thirdoperand of the compound equation can be exponential functions that varyas a function of distance from the midpoint of the magnetic solenoid 10along the central axis 20, such as to have counteracting additive andsubtractive effects.

As an example, the compound equation that defines the radius R of theconductive coil can be expressed as R=(First Operand)+(Second Operand).Specifically, one example of the compound equation can be expressed asfollows:

$\begin{matrix}{R = {\frac{\sqrt{\left\lbrack {{MinorAxis}^{2}*\left\lbrack {1 - \frac{\left( {Z - {Midpoint}} \right)^{2}}{{MajorAxis}^{2}}} \right\rbrack} \right\rbrack}}{2} + \frac{\left( {{{Z - {Midpoint}}}*A} \right)^{C}}{B}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

Where: MinorAxis is a minor axis dimension of the elliptical portion ofthe compound equation;

-   -   MajorAxis is a major axis dimension of the elliptical portion of        the compound equation;    -   Z is a distance along the central axis 20 relative to a first        end of the magnetic solenoid 10;    -   Midpoint is a distance along the central axis 20 from the first        end of the magnetic solenoid 10 to the midpoint of the magnetic        solenoid 10 along the central axis 20; and    -   A, B, and C are constants defining the exponential portion of        the compound equation.        Thus, the first operand is demonstrated in Equation 1 as        corresponding to the elliptical function and the second operand        is demonstrated in Equation 1 as corresponding to the        exponential function. It is to be understood that the magnitudes        of the constants MinorAxis, MajorAxis, A, B, and C can be set to        achieve desired dimensional and magnetic field characteristics        of the magnetic solenoid 10.

In the example of FIG. 1, the magnetic solenoid 10 is demonstrated assubstantially symmetrical about a plane at a midpoint of the magneticsolenoid 10 along the central axis 20, as demonstrated by the line 22around the periphery of the magnetic solenoid 10. The magnetic solenoid10 includes a central portion 24, a first end portion 26, and a secondend portion 28 that is opposite the first end portion 26. In the exampleof FIG. 1, the radius of the conductive coil in the central portion 24can be largely dominated by the first operand of Equation 1. Therefore,the radius of the conductive coil in the central portion 24 issubstantially elliptical, such that the minor axis of the definedellipse is coplanar with the plane of symmetry 22. Thus, in eachdirection along the central axis 20 away from the plane of symmetry 22,the radius of the conductive coil slopes inward toward the central axis20 in the central portion 24. In each of the end portions 26 and 28, theradius of the conductive coil is largely dominated by the second operandof Equation 1. Therefore, the radius of the conductive coil flaresradially outward from the central axis 20 at points along the centralaxis 20 away from the plane of symmetry 22 in the first and second endportions 26 and 28.

The first operand of Equation 1 (i.e., the elliptical function) can beset to provide a substantially uniform along-axis magnitude of themagnetic field within the inner volume 18 of the magnetic solenoid 10.Specifically, the magnetic field within the inner volume 18 of themagnetic solenoid 10 can be substantially uniform along the central axis20 based on the characteristics (e.g., MajorAxis and MinorAxis) of thefirst operand of Equation 1. In addition, the characteristics (e.g.,MajorAxis and MinorAxis) of the first operand of Equation 1 can also beset to define the physical dimensions (i.e., length and width) of themagnetic solenoid 10. Conversely, the second operand of Equation 1(i.e., the exponential function) can be set to provide a substantiallyuniform off-axis magnitude of the magnetic field within the inner volume18 of the magnetic solenoid 10. Specifically, the magnetic field withinthe inner volume 18 of the magnetic solenoid 10 can be substantiallyuniform at points in three-dimensional space that are radially separatedfrom the central axis 20 relative to the magnitude of the magnetic fieldat the central axis 20 based on the characteristics (e.g., constants A,B, and C) of the second operand of Equation 1. As an example, theconstant C can be an even number to set the second operand as aneven-order polynomial, can be an odd number to set the second operand asan odd-order polynomial, or can be any number, such as including afractional magnitude.

As a result of the geometry of the conductive coil of the magneticsolenoid 10, the magnetic solenoid 10 can be implemented to effectivelyprovide a substantially uniform magnetic field within the inner volume18 of the magnetic solenoid 10 for a variety of applications. As anexample, the magnetic solenoid 10 can achieve a magnetic fielduniformity of better than five parts per million in a spherical volumewith a radius equal to approximately one-eighth of a correspondingcylindrical radius and with a coil length-to-diameter ratio ofapproximately 7:11 as calculated for magnetic field coil behavior insidemagnetic shielding, such as that described above. As another example,the magnetic solenoid 10 can achieve a magnetic field uniformity ofbetter than one part per million in a spherical volume with a radiusequal to approximately one-eighth of a corresponding cylindrical radius,and with a coil length-to-diameter ratio of approximately 14:15 ascalculated for magnetic field coil behavior inside magnetic shielding,such as that described above.

In addition, the geometry of the conductive coil can be such that themagnetic solenoid 10 can be manufactured at a significantly smaller sizerelative to conventional magnetic solenoids. Specifically, conventionalmagnetic solenoids can typically be required to be manufactured at asignificantly greater length to achieve similar magnetic fielduniformity. As an example, the magnetic solenoid 10 can be manufacturedat less than 10 millimeters and still achieve a substantially uniformmagnetic field within the inner volume 18. By comparison, a conventionalstrictly cylindrical solenoid with an identical maximum radius wouldneed to be approximately 70 mm in length to achieve substantially thesame on-axis field uniformity in the same test volume while enclosed insimilar shielding. Furthermore, conventional magnetic solenoids that donot implement a cylindrical geometry (e.g., having an ellipticalgeometry) can have a constricted radius, such as at one or both ends,such that the size of an object which can be inserted into the innervolume of the conventional magnetic solenoid can be restrictive.However, the magnetic solenoid 10 can be designed to have only minordeviations in radius to still achieve the substantial uniformity in themagnetic field in the inner volume 18. As an example, an object insertedinto the inner volume 18 of the magnetic solenoid 10 may only be reducedin size by approximately 4% relative to an object that can be insertedinto a cylinder having a radius approximately equal to the maximumradius of the magnetic solenoid 10. Accordingly, the magnetic solenoid10 can be smaller and more versatile in application than conventionalmagnetic solenoids.

FIG. 2 illustrates an example of graphs 30 and 32 demonstratingalong-axis magnetic field in accordance with an aspect of the invention.FIG. 3 illustrates an example of graphs 40 and 42 demonstrating off-axismagnetic field in accordance with an aspect of the invention. In theexample of FIGS. 2 and 3, the graphs 30 and 40 can correspond to atypical, substantially cylindrical magnetic solenoid having a coilradius of approximately 7.5 mm and a coil length of approximately 14 mm.The graphs 32 and 42 can correspond to the magnetic solenoid 10 in theexample of FIG. 1, such as based on Equation 1, having substantiallysimilar dimensions as the typical, substantially cylindrical magneticsolenoid. For example, the magnetic solenoid 10 can have a MinorAxisvalue of approximately 7.5 mm, a MajorAxis value of approximately 34 mm,a Midpoint value of approximately 7 mm, and values for the constants A,B, and C of approximately 10³, 10⁸, and 4, respectively.

For the graphs 30, 32, 40, and 42 in the examples of FIGS. 2 and 3, atest volume of an approximately 2 mm sphere centered at the midpoint ofeach of the magnetic solenoids is chosen. The graphs 30 and 32 plotchanges in the magnetic field (Y-axis), in parts per million, versusdistance along the central axis (X-axis) of the respective magneticsolenoids. Similarly, the graphs 40 and 42 plot changes in the magneticfield (Y-axis), in parts per million, versus transverse distance fromthe central axis (X-axis) of the respective magnetic solenoids.

As demonstrated by the graph 32, the magnetic solenoid 10 hassignificantly greater along-axis magnetic field uniformity than thetypical, cylindrical magnetic solenoid demonstrated by the graph 30.Specifically, the graph 30 demonstrates variation in the along-axismagnetic field for the typical, substantially cylindrical magneticsolenoid of approximately 8000 parts per million relative to a variationof approximately 0.4 parts per million demonstrated by the graph 32 forthe magnetic solenoid 10. Similarly, as demonstrated by the graph 42,the magnetic solenoid 10 has significantly greater off-axis magneticfield uniformity than the typical, cylindrical magnetic solenoiddemonstrated by the graph 40. Specifically, the graph 40 demonstratesvariation in the off-axis magnetic field for the typical, substantiallycylindrical magnetic solenoid of approximately 4000 parts per millionrelative to a variation of approximately 0.8 parts per milliondemonstrated by the graph 42 for the magnetic solenoid 10.

Referring back to the example of FIG. 1, the compound equation thatdefines the radius of the conductive coil of the magnetic solenoid 10 isnot limited to Equation 1. As an example, the compound equation thatdefines the radius R of the conductive coil can be expressed as R=(FirstOperand)−(Second Operand)+(Third Operand). Specifically, another exampleof the compound equation can be expressed as follows:

$\begin{matrix}{R = {{MidpointRadius} - \frac{\left( {{{Z - {Midpoint}}}*A} \right)^{B}}{C} + \frac{\left( {{{Z - {Midpoint}}}*D} \right)^{E}}{F}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

Where: MidpointRadius is a radius at the midpoint of the magneticsolenoid 10 along the central axis 20;

-   -   Z is a distance along the central axis 20 relative to a first        end of the magnetic solenoid 10;    -   Midpoint is a distance along the central axis 20 from the first        end of the magnetic solenoid 10 to the midpoint of the magnetic        solenoid 10 along the central axis 20; and    -   A, B, C, D, E, and F are constants defining the respective        exponential portions of the compound equation.        Thus, the first operand is demonstrated in Equation 2 as        corresponding to the radius at the midpoint of the magnetic        solenoid 10 along the central axis 20, such that the first        operand can define the greatest value of the radius at the        central portion 24 of the magnetic solenoid 10. The second and        third operands are demonstrated in Equation 2 as corresponding        to exponential functions. As demonstrated in Equation 2, the        second operand is subtracted from first operand and the third        operand is added to the first and second operands. Therefore,        the second and third operands counteract to provide a varying        value of the radius along the central axis 20 of the magnetic        solenoid 10 from the midpoint to the respective first and second        end portions 26 and 28. It is to be understood that the        magnitudes of the constants A, B, C, D, E, and F can be set to        achieve desired dimensional and magnetic field characteristics        of the magnetic solenoid 10.

Based on the configuration of Equation 2, the first operand of Equation2 can provide an initial radial value, and the second and third operandsof Equation 2 can be defined (i.e., based on the constants A, B, C, D,E, and F) to provide substantially uniform along-axis and off-axismagnitudes of the magnetic field within the inner volume 18 of themagnetic solenoid 10. Specifically, the constants A, B, C, D, E, and Fin the second and third operands, respectively, can define both asubstantially rounded shape of the central portion 24 along the lengthof the magnetic solenoid to provide a substantially uniform magnitude ofthe magnetic field along the central axis 20. The second and thirdoperands of Equation 2 can also be defined to provide a substantiallyuniform off-axis magnitude of the magnetic field within the inner volume18 of the magnetic solenoid 10. Specifically, the constants A, B, C, D,E, and F in the second and third operands, respectively, can also definethe substantially flared shape of the respective end portions 26 and 28.Accordingly, the compound equation can be expressed in any of a varietyof ways to define the geometry of the conductive coil of the magneticsolenoid 10 to generate the substantially uniform magnetic field withinthe inner volume 18 of the magnetic solenoid 10.

FIG. 4 illustrates an example of a gyroscope system 50 in accordancewith an aspect of the invention. As an example, the gyroscope system 50can be configured as a nuclear magnetic resonance (NMR) gyroscopesystem, such as to detect rotational motion about an axis. For example,the gyroscope system 50 can be one of a plurality of gyroscope systems,such as implemented in air or spacecraft, to detect yaw, pitch, androll, respectively.

The gyroscope system 50 includes a gyroscope cell system 52. Thegyroscope cell 52 can include a glass case that can be filled with, forexample, an alkali metal vapor and/or at least one noble gas isotope. Inthe example of FIG. 4, the gyroscope cell 52 is substantially enclosedin an inner volume of a magnetic solenoid 54. As an example, themagnetic solenoid 54 can be configured substantially similar to themagnetic solenoid 10 in the example of FIG. 1. Specifically, themagnetic solenoid 54 can have a radius about a central axis that isdefined by a compound equation, such as Equation 1 that includes a firstoperand that is a substantially circular or elliptical function and asecond operand that is an exponential function (e.g., a polynomial), orsuch as Equation 2 that includes a first operand that defines the radiusat the center of the magnetic solenoid 54 and second and third operandsthat are each exponential functions. Therefore, the magnetic solenoid 54can be configured to generate a substantially uniform magnetic fieldB_(E) throughout the inner volume of the magnetic solenoid 54.

In response to the substantially uniform magnetic field B_(E), thealkali metal vapor and the noble gas isotope(s), in the gyroscope cell52 can precess relative to the axis of the gyroscope cell 52. Thegyroscope system 50 also includes an opto-electronics system 56. Theopto-electronics system 56 can be configured to optically pump thealkali metal vapor in the gyroscope cell 52 to align the spin of thealkali metal vapor with the applied magnetic field. In response, due toa spin-exchange process, any noble gas isotopes also present in the cellare also spin-aligned to the pump light beam. A detection beam of lightwith a directional component normal to the pump direction can bemodulated in response to the alignment of the alkali metal vaporrelative to the detection light. The modulation of the detection lightcan be a function of the precession of the alkali metal vapor asmodified by the precession of any noble gas isotopes present. Thismodulation can be detected by a photodetector, such as included in theopto-electronics system 56. Accordingly, changes in the precession ratesof the alkali metal vapor, and by extension any noble gas isotopespresent, as detected by the modulated optical signal, can be detectedand processed to determine changes in the orientation of the gyroscopecell 52 that correspond to rotational motion.

It is to be understood that the gyroscope system 50 is demonstratedsimplistically. As such, additional components and details have beenomitted from the example of FIG. 4 and the associated description. Inaddition, it is also to be understood that a magnetic solenoid such asthe magnetic solenoid 10 in the example of FIG. 1 is not intended to belimited to use in a gyroscope system, such as the gyroscope system 50 inthe example of FIG. 4. Therefore, the magnetic solenoid 10 can beimplemented in any of a variety of applications that may require amagnetic field, particularly a substantially uniform magnetic field.

What have been described above are examples of the present invention. Itis, of course, not possible to describe every conceivable combination ofcomponents or methodologies for purposes of describing the presentinvention, but one of ordinary skill in the art will recognize that manyfurther combinations and permutations of the present invention arepossible. Accordingly, the present invention is intended to embrace allsuch alterations, modifications and variations that fall within thespirit and scope of the appended claims.

1. A magnetic solenoid comprising an elongated sidewall that extendsbetween spaced apart ends, the elongated sidewall surrounding a centralaxis that extends longitudinally along the sidewall, the elongatedsidewall having a radius that is defined by a compound equation thatvaries the radius as a function of position along the central axis. 2.The magnetic solenoid of claim 1, wherein the compound equationcomprises a first operand that affects along-axis uniformity of thesubstantially uniform magnetic field and a second operand that affectsoff-axis uniformity of the substantially uniform magnetic field.
 3. Themagnetic solenoid of claim 2, wherein the first operand comprises asubstantially elliptical function with respect to the radius and havinga minor axis bisected by the central axis, and wherein the secondoperand comprises an exponential function that causes the radius toflare outward from the central axis at the spaced apart ends.
 4. Themagnetic solenoid of claim 1, wherein the magnetic solenoid issubstantially symmetrical with respect to a plane that is normal to thecentral axis and which intersects a midpoint of the magnetic solenoidalong the central axis.
 5. The magnetic solenoid of claim 1, wherein thecompound equation comprises: a first operand that defines the radius ata midpoint of the magnetic solenoid along the central axis; a secondoperand that is an exponential function that is subtracted from firstoperand; and a third operand that is an exponential function that isadded to the first operand, the second and third operands counteractingeach other to vary the radius along a length of the magnetic solenoidfrom the midpoint.
 6. The magnetic solenoid of claim 1, wherein thesubstantially uniform magnetic field has a substantially uniformmagnitude with respect to each point in three-dimensional space withinthe inner volume of the magnetic solenoid.
 7. The magnetic solenoid ofclaim 1, wherein the radius R is defined by:${R = {\frac{\sqrt{\left\lbrack {{MinorAxis}^{2}*\left\lbrack {1 - \frac{\left( {Z - {Midpoint}} \right)^{2}}{{MajorAxis}^{2}}} \right\rbrack} \right\rbrack}}{2} + \frac{\left( {{{Z - {Midpoint}}}*A} \right)^{C}}{B}}},$where: MinorAxis is a minor axis dimension of an elliptical portion ofthe compound equation; MajorAxis is a major axis dimension of theelliptical portion of the compound equation; Z is a distance along thecentral axis relative to a first end of the magnetic solenoid; Midpointis a distance along the central axis from the first end of the magneticsolenoid to a midpoint of the magnetic solenoid along the central axis;and A, B, and C are constants defining an exponential portion of thecompound equation.
 8. The magnetic solenoid of claim 1, wherein theradius R is defined by:${R = {{MidpointRadius} - \frac{\left( {{{Z - {Midpoint}}}*A} \right)^{B}}{C} + \frac{\left( {{{Z - {Midpoint}}}*D} \right)^{E}}{F}}},$where: MidpointRadius is a radius at a midpoint of the magnetic solenoidalong the central axis; Z is a distance along the central axis relativeto a first end of the magnetic solenoid; Midpoint is a distance alongthe central axis from the first end of the magnetic solenoid to themidpoint of the magnetic solenoid along the central axis; and A, B, C,D, E, and F are constants defining the respective exponential portionsof the compound equation.
 9. A gyroscope system comprising the magneticsolenoid of claim 1, the gyroscope system comprising a gyroscope cellthat is enclosed within the inner volume of the magnetic solenoid.
 10. Amagnetic solenoid comprising an elongated sidewall that extends betweenspaced apart ends, the elongated sidewall surrounding a central axisthat extends longitudinally along the sidewall, the elongated sidewallhaving a radius that is defined by a compound equation having a firstoperand that affects along-axis uniformity of a substantially uniformmagnetic field and a second operand that affects off-axis uniformity ofthe substantially uniform magnetic field, such that the substantiallyuniform magnetic field has a substantially uniform magnitude anddirection with respect to each point in three-dimensional space withinan inner volume that is enclosed by the conductor coil.
 11. The magneticsolenoid of claim 10, wherein the first operand comprises asubstantially elliptical function with respect to the radius of theconductor coil and having a minor axis bisected by the central axis, andwherein the second operand comprises an exponential function that causesthe radius to flare outward from the central axis at the spaced apartends.
 12. The magnetic solenoid of claim 10, wherein the magneticsolenoid is substantially symmetrical with respect to a plane that isnormal to the central axis and which intersects a midpoint of thecentral axis.
 13. The magnetic solenoid of claim 10, wherein the firstoperand is a substantially elliptical function that is summed with thesecond operand to define the radius.
 14. The magnetic solenoid of claim13, wherein the first operand is described by:$\frac{\sqrt{\left\lbrack {{MinorAxis}^{2}*\left\lbrack {1 - \frac{\left( {Z - {Midpoint}} \right)^{2}}{{MajorAxis}^{2}}} \right\rbrack} \right\rbrack}}{2}$Where: MinorAxis is a minor axis dimension of the elliptical function ofthe compound equation; MajorAxis is a major axis dimension of theelliptical function of the compound equation; Z is a distance along thecentral axis relative to a first end of the magnetic solenoid; andMidpoint is a distance along the central axis from the first end of themagnetic solenoid to a midpoint of the magnetic solenoid along thecentral axis.
 15. The magnetic solenoid of claim 10, wherein the secondoperand is an exponential function that is summed with the first operandto define the radius.
 16. The magnetic solenoid of claim 15, wherein thesecond operand is described by:$\frac{\left( {{{Z - {Midpoint}}}*A} \right)^{C}}{B}$ Where: Z is adistance along the central axis relative to a first end of the magneticsolenoid; Midpoint is a distance along the central axis from the firstend of the magnetic solenoid to a midpoint of the magnetic solenoidalong the central axis; and A, B, and C are constants defining theexponential function of the compound equation.
 17. A magnetic solenoidthat is configured to provide a substantially uniform magnetic field inan inner volume that is enclosed by the magnetic solenoid, the magneticsolenoid comprising: a central portion in which a radius of the magneticsolenoid about a central axis is substantially elliptical, such that anelliptical minor axis occupies a plane that is normal to the centralaxis; a first end portion in which the radius of the magnetic solenoidflares outward from the central axis; and a second end portion oppositethe first end portion in which the radius of the magnetic solenoidflares outward from the central axis.
 18. The magnetic solenoid of claim17, wherein the radius of the conductor coil in each of the central,first end, and second end portions is defined by a compound equationthat varies the radius as a function of position along the central axis.19. The magnetic solenoid of claim 18, wherein the compound equationcomprises: a first operand that is a substantially elliptical functionwith respect to the radius of the conductor coil that affects along-axisuniformity of the substantially uniform magnetic field; and a secondoperand that is an exponential function that affects off-axis uniformityof the substantially uniform magnetic field.
 20. The magnetic solenoidof claim 17, wherein the magnetic solenoid is symmetrical with respectto a plane that is normal to the central axis and which intersects themidpoint of the central axis.